The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date.
How much should be invested now (the present value) to have an amount of $10,000, 3 years from now, if the amount is invested at an interest rate of 4% per year, compounded semiannually. (Round your answer up to the next cent.)
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To calculate the present value (PV) of an amount of money, given the future value (FV), interest rate, time period, and compounding frequency, we use the following formula:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $PV$ is the present value (what we want to find),
- $FV = 10,000$ is the future value,
- $r = 0.04$ is the annual interest rate (4%),
- $n = 2$ is the number of compounding periods per year (semiannually),
- $t = 3$ is the number of years.

Let's calculate the present value:

\[
PV = \frac{10,000}{(1 + \frac{0.04}{2})^{2 \times 3}}
\] 

I will now compute this value.The present value (PV) of $10,000 to be received in 3 years, at an interest rate of 4% per year compounded semiannually, is **$8,879.71**.

Would you like further details on this calculation, or have any questions?

Here are some related questions to consider:
1. How does the frequency of compounding affect the present value?
2. What would the present value be if the interest rate increased to 5%?
3. How does continuous compounding differ from semiannual compounding in this scenario?
4. What happens to the present value if the time period is extended to 5 years?
5. How does inflation affect the real value of money over time?

**Tip**: The higher the frequency of compounding, the lower the present value will be for the same interest rate and future value.

How much should be invested now (the present value) to have an amount of $10,000, 3 years from now, if the amount is invested at an interest rate of 4% per year, compounded semiannually?

Learn how to calculate the present value of $10,000 to be received in 3 years, with an interest rate of 4% per year compounded semiannually. This step-by-step guide explains how to use the present value formula and includes examples for financial calculations. Suitable for Grades 10-12.

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